[Autoplectic]

Summary

A channel p(g | s+, s-) that renders the forward and reverse causal states conditionally independent need not be the hidden state of a generator. The HMM sofic builds from it (the channel applied at both ends of every bidirectional edge) generates the process, with G jointly distributed with past and future as the channel prescribes, if and only if

• I[G_0 X_0 : S-_1 | G_1] = 0 (the next state screens what was emitted from the reverse causal state), and
• I[S+_0 : X_0 G_1 | G_0] = 0 (its time reverse).

Previously the exact/Wyner constructors ignored these, checked reproduction afterwards, and silently returned the functional model under the unconstrained value. The functional model itself was returned without any check and can fail too.

Changes

• minimal_generative_model / wyner_generative_model optimize subject to both conditions (new sofic/generators/lifted_common_information.py): SLSQP with analytic constraint Jacobians, continuation from the unconstrained problem, starts at S+, S-, (S+, S-), the dit optimum and random channels, then refinement on the channel's support. The reported value is the model's own; the exact value upper-bounds the generative complexity. The fallback is removed and models carry markov_residual (largest constraint information, bits).
• functional_generative_model searches deterministic partitions under the same conditions (exhaustive up to nine joint states, greedy merging from (S+, S-) beyond), so it always reproduces the process. dit's partition is kept when it already satisfies them (it is then optimal).
• New measures BidirectionalEpsilonMachine.wyner_common_information() / exact_common_information() return the unconstrained C[S+:S-], G[S+:S-] (= the common informations of past and future).
• generative_gap(measure="exact" | "wyner") (also BidirectionalEpsilonMachine.generative_gap()) returns GenerativeGap(lower, upper, gap, model): the unconstrained common information vs. its value over generator states. For the exact measure G[S+:S-] <= C_g <= H[G], so gap bounds C_g - G[S+:S-]. On the (2,∞) process: exact 1.406 → 1.5 (gap 0.094), Wyner 1.129 → 1.311 (gap 0.18); zero for first-order Markov processes.
• Bug fix: dit numbers each variable's symbols by first appearance, so the optimum was read with the reverse-state axis permuted whenever the support listed a higher-indexed reverse state first. The resulting channels did not separate S+ from S- (this is why the golden mean appeared to fail and fell back).

Motivating example

Every 1 followed by at least two 0s (3-state ε-machine, one ½/½ branch, second-order Markov) is the simplest case where the conditions bind. Its unconstrained exact optimum is a deterministic W with H[W] = 1.406; one value of W ("past ends in 10, future starts with 01") can only be preceded by a value that does not know the past ended in 10, so no HMM has W as its state. Constrained values: exact 1.5, Wyner 1.311 (vs 1.129 unconstrained). For first-order Markov processes the constrained and unconstrained values coincide.

Breaking changes

Default behavior of the three constructors and generative_complexity() changes (see CHANGELOG). There is no unconstrained-model mode; use the new measures for the unconstrained values.

Tests

• New tests/test_lifted_common_information.py: golden mean and binary Markov chains vs closed-form C_g, Odd process (interior Wyner generator), the (2,∞) process, functional model always reproducing (exhaustive and greedy paths), set-partition enumeration vs Bell numbers, Jacobian vs finite differences, Hypothesis bounds on Markov chains.
• Regression test for the dit symbol-order alignment.
• Full suite: 2019 passed, 15 skipped; ruff clean. Sphinx build not run locally.